Synthetic approach to the Quillen model structure on topological spaces

TL;DR

This paper develops an axiomatic framework for Quillen's model structure on topological spaces, extending its applicability to broader categories like $ riangle$-generated spaces, pseudotopological spaces, and locales.

Contribution

It introduces an axiomatic approach to Quillen's construction, enabling the extension of the model structure to new categories such as locales.

Findings

Axiomatization of Quillen's model structure

Extension to $ riangle$-generated and pseudotopological spaces

Construction of a model structure on locales

Abstract

We provide an axiomatic treatment of Quillen's construction of the model structure on topological spaces to make it applicable to a wider range of settings, including $\Delta$-generated spaces and pseudotopological spaces. We use this axiomatization to construct a model structure on the category of locales.